NBSIR 74-549^0

Nucleus-Nucleus Collisions at

Very High Energies

A. Bia/as

Max-Planck Institute for Physics and Astrophysics

Munchen, Germany

W. Czyz

Institute of Nuclear Physics

Cracow, Poland

L. C. Maximon

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

August 1974

Work supported by:

NBS Special Foreign Currency Program, Grant No. (G) 173.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 74-549

NUCLEUS-NUCLEUS COLLISIONS AT

VERY HIGH ENERGIES

A. Bia/as

Max-Planck Institute for Physics and Astrophysics

Munchen, Germany

W. Czyz

Institute of Nuclear Physics

Cracow, Poland

L. C. Maximon

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

August 1974

Work supported by:

NBS Special Foreign Currency Program, Grant No. (G) 173.

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

nucleus-nucleus collisions at very high energies

Ao Bialas

Max-Planck Institute for Physics and Astrophysics, Mttnchen

W. Czyz

Institute of Nuclear Physics, Cracow

L.C, Maximon

National Bureau of Standards, Washington D.C.

I. Introduction

It has been stressed many times that nucleus-nucleus

collisions at high energies are worth investigating "both

theoretically and experimentally [l-6^] because virtually

nothing is known about this exotic field and, at the same

time, the theoretical models one may envisage give several

simple and definite predictions. The present note discusses

one such model which was originally proposed for hadron-nucleus

scattering f 7] but can be straightforwardly generalized to

nucleus-nucleus collisions. The main purpose of this note is

to spell out this generalization and to give the formulae for

the total cross sections, diffractive production cross sections,

and inclusive single particle distributions in multiparticle

production in nucleus-nucleus collisions. One of the important

conclusions of this analysis is that the measurements of the

relations between various total cross sections proposed in

refs, £3-6] could test the model of ref . £7] •

^"work sponsored by NBS Special Foreign Currency Program, Grant No.(G)173.

*)On leave of absence from Institute of Physics, Jagellonian Univ.,

Cracow, Poland, and Institute of Nuclear Physics, Cracow, Poland.
- 2

2 * The__model •

We shall work in the cm, system of the two nuclei

containing A and B nucleons. The S matrix for such ^collision

is, according to the model of ref. [~7J »

the

where j:* is the S matrix for /j-th nucleon from A colliding

with jf-th nucleon from B. As in 7 » the "aosorptive" part,

A Pjgj which describes both elastic scattering and diffractive

production is generated by the driving term T, responsible

^ / ~ t \ for non-diffractive multiple production processes ( ~ 'j^ / and the unitarity of V. gives

J*

h

' JS V- ' V • C2>2;

Again, iollowing the approach of ref, [?]» we accept that

the production processes ^in fact all possible processes^ are

completely coherent over the two nuclei and this fact is

incorporated into our description in a relativistically

invariant manner by making the operators of Eqs. (2.l) and 0^*^)

dependent only on the transverse degrees of freedom . In the

present simple version we shall use only the impact parameter
- 2 -

and the transverse distances of the individual nucleons from

the axis of cylindrical symmetry of the collision as these

transverse degrees of freedom.

One should also keep in mind that, henceforth, we shall

deal only with sum rules in which the momentum transfer

between the two nuclei ( more precisely: between the two groups

of particles which emerge from the collision and go to the

left and to the right in the cm. system) is integrated over

and the sum performed over all possible excitations (of nuclei

and of their components: nucleons) • Gross sections for

specific processes are obtained, in the present approach,

through the introduction of sutable projection operators into

such sums and integrations. Note that in this approach one

completely avoids dealing with the longitudinal momentum

transfers in the process of multiple collisions: for example,

as can be seen in Sec. Con inclusive cross sections, the

dependence on the longitudinal momenta comes only through

the final expression, (2.6) , where they are accounted for

in the "elementary" hadron-nucleon inclusive cross section,

45; (W •

Assuming the same properties of the individual

matrices as in ref. [7 J and going through algebra analogous

to that used in the case of hadron-nucleus collisions [7 J , we readily obtain:

A. Elastic scattering and coherent diffractive production . The amplitude for both processes at the impact parameter
- 4 -

b reads (to lowest order in diffractive production)

A A

le,

>

(2.3)

0 £ (A)j <Sf

In this formula the absorptive part of the matrix was

split into the elastic and diffractive inelastic parts

where y.'n is diagonal ( this is the "profile" of the Glauber

\

J model! and the off-diagonal correction responsible for

diffractive production is assumed to be small. In ref . £7]

such a form for V. was explicitely obtained from the

ft

bremsstrahlung model of multi-particle production [bj .

It was also suggested by the general considerations ox ref. [9J

In Eq. (2,5) the states are labelled by the nuclear wave

functions of both nuclei Cp^j (ft ^ yan<3 ^y the produced
- 5 -

particles, schematically denoted fh) • The first part of the

r.h.s. of (2.3) gives the well known nucleus-nucleus high

r

-i the

energy elastic scattering amplitude [1J • Note that/second

part of the r.h.s. of (2.3] , which gives diffractive

production, is proportional to AB^ which means that the cross

2 2 sections have a A B dependence. Hence it would seem that

diffractive production in nucleus-nucleus collisions may

contribute more significantly to, e.g., one particle inclusive

distributions (see below) than in the case of hadron-nucleus

collisions, where B = lo

B. The total cross sections . Similarly, as in [7] , after

employing closure 2 E K) $^fXtfty** ] ^ I = I and integrating over the momentum transfer between the two

nuclei, we obtain the following formula for the total nucleusnucleus cross section:

(2.4)

\

where we employed unitarity: 5^ n S^g — I • Again, as in

£7] * the total cross section is, to zeroth order in <£ ,
the same as that given by the Glauber model refs* ("l,5>4- I , and all the relations between the total cross sections

proposed in refs. [3,4,5 >6jf » when compared with experiment,

will provide a test for the multiparticle production model

proposed in re£ [ 7J and used here.

Go Inclusive single particle distribution in multiparticle

production in nucleus-nucleus collisions . We again follow

the same algebra as in £7 J

:

(2.5)

The commutation relations C a

^p; ) ; ^jj were worked

out in for the case of the hadronic bremsstrahlung model

When we employ them, we again obtain the result that the

"diagonal" terms give the leading contribution:

(2.6)
- 7 -

and the non-diagonal contributions appear to be small.

The same qualifications made regarding Eq. (ll) of

ref. \J?\ > viz., that corrections are to be expected in the

very forward direction, can be made here. However, we would

like to emphasize the following point. In performing the

algebraic operations which lead to (2.6^ or Eq. ^ll) of

ref. £7] one neglects energy-momentum conservation in Ji\ , which results in suppressing diffractive production . Hence

neither Eq. (2.6j nor Eq. (ll) of ref. contain

contributions to coming from diffractive production*,

In the case of ha dron-nucleus interactions this simplification

may still be not too bad, but, as has already been pointed

out in this note, in the case of nucleus-nucleus collisions

it is more serious. It would seem that one could estimate

the relevant corrections starting directly from Eq. (2.3,) • In any case one should expect the relation (2 .6] to fail in

the extreme forward and backward directions, which are

populated by products of diffractive processes.

In concluding, one may say that one can literally take

over the model of ref. £7] a n<3 extend it trivially to the

the

case of nucleus-nucleus scattering gust as/ Glauber model for

hadron-nucleus elastic scattering can be extended to

nucleus-nucleus elastic scattering.
- 8 -

References

1. W. Czyz and L.C. Maximon, Ann. Phys. 52, 59 (JL969) §

Z. H.H. Heckman, High Energy Heavy Ions: A New Area for

Physics Research, 5th Int. Conf • on High-Energy Physics

and Nuclear Structure, Uppsala, June 1973*

3. P.M. Pishbane and J.S. Trefil, Phys. Rev. Letters 32, 396

(.1974),

4 0 V. Franco, Phys. Revo Letters ^2, 911 C-WM

5. S. Barshay, G.B. Dover and J. P. Vary, The Validity of the

Factorization Hypothesis for Nucleus-Nucleus Cross Sections

at High Energies, Brookhaven preprint BNL 15811, April, 1974.

6. S. Barshay, G.B. Dover and J. P. Vary, Nucleus-Nucleus

Cross Sections and the Validity of the Factorization

Hypothesis at Intermediate and High Energies, Brookhaven

preprint BNL 18972, May, 1974.

7. A. Bialas and W. Czyz, Short-time behavior of hadronic

interactions and the hadron-nucleus production processes,

Max-Planck Institute for Physics and Astrophysics preprint,

to be published in Physics Letters.

do A. Bialas and A. Kotahski, Acta Physica Polonica B4, 659

(1973) o

9. W. Czyz, Phys. Rev. D8, 3219 (1973) *
NBS-114A IHF. V. 7-73)

U.S. D€PT. OF" COMM.

BIBLIOGRAPHIC DATA

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1. PUBLICATION OR REPORT NO. NBSIR 74-549

2. Gov't Accession

No.

3. Recipient's Acccsr.iun No.

4. TITLE AND SUBTITLE

Nucleus-Nucleus Collisions at Very High Energies

5. Publication Dace August 1974

6. Performing Organization Code

7. AUTHOR(S)

A. Bialas, W. Czyz , and L. C. Maximon

8. Performing Organ. Report No. NBSIR 74-549

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2400103

11. Contract/Grant No.

12. Sponsoring Organization Name and Complece Address (Street, City, State, ZIP)

Same as No. 9

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary ol most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

A model for high energy nucleus-nucleus collisions is presented. Formulae are given for the total cross sections, diffractive productive cross sections, and

inclusive- single particle distributions in multiparticle production in nucleusnucleus collisions. It is noted that measurement of the relations between a few

nucleus-nucleus total cross sections could be used to test this model.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Diffractive production cross sections; glauber model; high eneVgy hadron interactions;

inclusive single particle distributions; nucleus-nucleus collisions; total hadron

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